Extreme value theory and market risk

Abstract

In this dissertation we shall be modelling extreme market returns using Extreme Value Theory under the assumption, that these returns are independent and identically distributed random variables. These extreme events shall be modelled using both the Block Maxima method and the Peak over Threshold method, which are the two approaches of Extreme Value theory. The Block Maxima method models maxima taken every time period, using the Generalized Extreme Value distribution, while the Peak over Threshold method models exceedances taken above some carefully chosen threshold, using the Generalized Pareto distribution. From these models, we shall estimate the return levels for future losses, such as 10 year losses as in the case of this dissertation. Furthermore, we shall describe two risk measures known as Value at Risk and Expected Shortfall, and following the models obtained using Extreme value theory, we shall estimate these risk measures using the maximum likelihood estimated parameters of the fitted distributions. Value at Risk provides a level of loss such that with some probability a , this level is not exceeded in the next time period, while Expected Shortfall on the other hand, is the expected loss value of the next time period, given that the Value at Risk level is exceeded. The results for these estimations shall determine whether it is reasonable to consider the method as a suitable method to estimate return levels, while the Peak over Threshold method as being more suitable for estimating risk measures. Furthermore, we shall compare the results of the Value at Risk for the Extreme Value approaches, with Value at Risk levels obtained under the assumption that returns are normally distributed, as proposed by the Basel Accords. From this, we can conclude whether the use of Extreme Value Theory in estimating market risk, is in fact a better approach then the previously adopted normality distributed returns approach.